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In this paper, we study the Cauchy problem of a two-phase flow system consisting of the compressible isothermal Euler equations and the incompressible Navier-Stokes equations coupled through the drag force, which can be formally derived from the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations. When the initial data is a small perturbation around an equilibrium state, we prove the global well-posedness of the classical solutions to this system and show the solutions tends to the equilibrium state as time goes to infinity. In order to resolve the main difficulty arising from the pressure term of the incompressible Navier-Stokes equations, we properly use the Hodge decomposition, spectral analysis, and energy method to obtain the L 2 time decay rates of the solution when the initial perturbation belongs to L 1 space. Furthermore, we show that the above time decay rates are optimal. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the three‐dimensional isentropic compressible Navier–Stokes/Allen–Cahn system, which models the phase transitions in two‐component patterns interacting with a compressible fluid. Under the assumption that the initial perturbation is small and decays spatially, we establish the global existence and the pointwise behavior of strong solutions to this nonconserved system. To deal with the source terms involving the phase variable, we employ the Green's function and space‐time weighted estimates. The analysis shows that the phase variable mainly contains the diffusion wave with exponential decaying amplitude over time, and consequently the density and momentum of the compressible fluid adhere to a generalized Huygens principle. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate [Math Processing Error] (1 + t) − 1 2 in [Math Processing Error] L ∞ norm. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence and large time behavior of the classical solution for the non-isentropic compressible Navier-Stokes-Allen-Cahn (NSAC) system describing the mixture motion of a two-phase immiscible viscous compressible heat-conducting flow in R 3. We first establish the global well-posedness of the Cauchy problem to the non-isentropic compressible NSAC system when the initial data is a small regular perturbation near the constant state. Then by a delicate spectral analysis and the damping structure for the perturbation equation of the phase field, we prove that the phase field decays to the phase separation state at an exponential rate while the density, velocity and the temperature converge to the constant state at the optimal time rate (1 + t) − 3 / 4 in L 2 norm. Furthermore, for well-chosen initial data, we also establish the lower bounds of time decay rate. [ABSTRACT FROM AUTHOR]